Explicit or general rule

The rule for the terms is \(a_n=5^n\). Substituting \(n=4\), we get \(a_4=5^4=5\times5\times5\times5=625\). Option 125 is the value of \(5^3\), so it is a close but incorrect distractor. Exam tip: in an explicit rule, substitute the required term number directly for \(n\).

The numerator is (2n-1) and the denominator is (n+1), so (a_n=\frac{2n-1}{n+1}). In a fraction sequence form rules for both parts separately.

(a_5-a_2=9) and there are three gaps, so (d=3), then (a_n=3n+1). From two given terms first find the common difference.

Substitute \(n=1,2,3\) into the rule: \(a_1=1^2+3(1)-1=3\), \(a_2=2^2+3(2)-1=9\), and \(a_3=3^2+3(3)-1=17\). Thus, the first three terms are \((3, 9, 17)\), so option B is correct. In option A, the second term is 9, but the first and third terms do not follow the rule. Exam tip: for a general-term question, begin by substituting \(n=1\).

Given \(a_n=kn+2\). Substituting \(n=7\) gives \(a_7=7k+2\). Thus, \(7k+2=37\), so \(7k=35\) and \(k=5\). If \(k=6\), then \(a_7=44\), not 37. Exam tip: substitute the given term number into the general term to find an unknown constant.

The first term is (4) and the difference is (6), so (a_n=4+(n-1)6=6n-2). Do not forget the first term while forming a rule from words.

The correct rule is \(a_n=n^2+n+1\). Substituting \(n=1,2,3,4\) gives \(3,7,13,21\), respectively. In option A, the second term is \(6\), so it does not match the sequence. In an exam, verify a rule using at least the first two or three terms.

The general rule gives the value of the term directly when its position is known: \\(a_n=\\frac{n}{n+2}\\). We need to find the position n for which the term equals \\(\\frac{5}{7}\\). Thus, substitute the required value and solve the resulting equation. Since a term number is normally a positive integer, the valid solution must be checked as a whole-number position.

Set \\(\\frac{n}{n+2}=\\frac{5}{7}\\). Cross-multiplication gives \\(7n=5(n+2)\\), so \\(7n=5n+10\\). Subtracting \\(5n\\) from both sides gives \\(2n=10\\), hence \\(n=5\\). Checking gives \\(a_5=\\frac{5}{5+2}=\\frac57\\). Therefore, the fifth term, option D, is correct.

Given \(a_n=2n^2+3\), \(a_3=2(3)^2+3=18+3=21\) and \(a_4=2(4)^2+3=32+3=35\). Therefore, \(a_3+a_4=21+35=56\). The value 52 may result from not including the constant term \(+3\) correctly in both terms. Exam tip: substitute the value of \(n\) and calculate each term separately before adding.

Taking the term number as \(n=1,2,3,4,\ldots\), the rule \(a_n=n^2-1\) gives \(1^2-1=0\), \(2^2-1=3\), \(3^2-1=8\), and \(4^2-1=15\). Hence, option B is correct. Option C gives \(0,2,4,6,\ldots\), which is a linear pattern and does not match the given sequence. Exam tip: substitute \(n=1,2,3,4\) to verify an explicit rule quickly.

Substitute \(n=1,2,3,4\) respectively: \(a_1=2^1+1=3\), \(a_2=2^2+2=6\), \(a_3=2^3+3=11\), and \(a_4=2^4+4=20\). Hence, the correct sequence is \((3,6,11,20)\). Option C has the correct first and fourth terms, but its second and third calculations are incorrect. Exam tip: evaluate \(2^n\) first, then add \(n\).

The magnitude is (3,5,7,9,\ldots) and the signs start negative and alternate, so (a_n=(-1)^n(2n+1)). In alternating signs always check the sign of the first term.

Given \(n^2+2n=80\), we get \(n^2+2n-80=0\). Factoring gives \((n+10)(n-8)=0\), so \(n=-10\) or \(n=8\). Since a sequence index \(n\) is a positive integer, \(n=8\) is valid. Option 9 is a close distractor, but \(a_9=81+18=99\), not 80. Exam tip: for term-position questions, check whether a negative root is allowed before selecting it.

For \(a_n=n^2-2n+2\), substituting \(n=1,2,3,4\) gives \(1,2,5,10\), respectively. Hence, it is the required general term. Option B gives the first term as \(2\), so it cannot represent the sequence. In an exam, verify a proposed general term by substituting the first few values of \(n\).

The governing concept is an explicit or general rule: a formula must produce the term directly from its position n. Rewrite the sequence as 2³, 3³, 4³, 5³, ... . The first displayed term corresponds to n = 1, so its cube base is n + 1; therefore the rule is aₙ = (n + 1)³. Checking confirms it: for n = 1, a₁ = 2³ = 8; for n = 2, a₂ = 3³ = 27; for n = 3, a₃ = 4³ = 64; and for n = 4, a₄ = 5³ = 125. Thus option B is correct. Option A starts with 1³, option C gives 2 at n = 1, and option D starts with 3³, so those alternatives do not match the sequence.

The correct rule is \(a_n=n^2+5n\). Substituting \(n=1,2,3,4\) gives \(6,14,24,36\), respectively. Also, the first differences are \(8,10,12\), whose differences are \(2\); hence the sequence should have a quadratic rule. Option B gives the first term as 6, but for \(n=2\) it gives 12, not 14. Exam tip: test a proposed rule with at least the first three values, \(n=1,2,3\).

In a linear rule the coefficient of (n) is (-3), so the common difference is (-3). Read the coefficient of (a_n) carefully.

Given \(a_n=4n^2-1\), \(a_5=4(5)^2-1=100-1=99\) and \(a_1=4(1)^2-1=3\). Hence, \(a_5+a_1=99+3=102\). Option 100 is only \(4\times 5^2\); it ignores both the \(-1\) and \(a_1\). Exam tip: Substitute each required value of \(n\) separately before adding the terms.

The general term is (a_n=50-5n), and (50-5n=0) gives (n=10). Even in a decreasing sequence the position remains natural.

Substitute n=1, 2, and 3 to obtain the first three terms. a₁=2(1)²+2(1)+1=5, a₂=2(2)²+2(2)+1=13, and a₃=2(3)²+2(3)+1=25. Hence, the correct sequence is (5, 13, 25). Option C has the correct first term, but for n=2 the value is 13, not 11. Exam tip: evaluate the squared term first and substitute each value of n separately.